Tensor product
Tensor product of two unitary modules
The tensor product of two unitary modules and
over an associative commutative ring
with a unit is the
-module
together with an
-bilinear mapping
![]() |
which is universal in the following sense: For any -bilinear mapping
, where
is an arbitrary
-module, there is a unique
-linear mapping
such that
![]() |
The tensor product is uniquely defined up to a natural isomorphism. It always exists and can be constructed as the quotient module of the free -module
generated by the set
modulo the
-submodule
generated by the elements of the form
![]() |
![]() |
![]() |
![]() |
![]() |
then . If one gives up the requirement of commutativity of
, a construction close to the one described above allows one to form from a right
-module
and a left
-module
an Abelian group
, also called the tensor product of these modules [1]. In what follows
will be assumed to be commutative.
The tensor product has the following properties:
![]() |
![]() |
![]() |
![]() |
for any -modules
,
and
.
If and
are bases of the free
-modules
and
, then
is a basis of the module
. In particular,
![]() |
if the are free finitely-generated modules (for instance, finite-dimensional vector spaces over a field
). The tensor product of cyclic
-modules is computed by the formula
![]() |
where and
are ideals in
.
One also defines the tensor product of arbitrary (not necessarily finite) families of -modules. The tensor product
![]() |
is called the -th tensor power of the
-module
; its elements are the contravariant tensors (cf. Tensor on a vector space) of degree
on
.
To any pair of homomorphisms of -modules
,
, corresponds their tensor product
, which is a homomorphism of
-modules
and is defined by the formula
![]() |
This operation can also be extended to arbitrary families of homomorphisms and has functorial properties (see Module). It defines a homomorphism of -modules
![]() |
![]() |
which is an isomorphism if all the and
are free and finitely generated.
Comments
An important interpretation of the tensor product in (theoretical) physics is as follows. Often the states of an object, say, a particle, are defined as the vector space over
of all complex linear combinations of a set of pure states
,
. Let the pure states of a second similar object be
,
, yielding a second vector space
. Then the pure states of the ordered pair of objects are all pairs
and the space of states of this ordered pair is the tensor product
.
Tensor product of two algebras
The tensor product of two algebras and
over an associative commutative ring
with a unit is the algebra
over
which is obtained by introducing on the tensor product
of
-modules a multiplication according to the formula
![]() |
This definition can be extended to the case of an arbitrary family of factors. The tensor product is associative and commutative and contains a unit if both algebras
have a unit. If
and
are algebras with a unit over the field
, then
and
are subalgebras of
which are isomorphic to
and
and commute elementwise. Conversely, let
be an algebra with a unit over the field
, and let
and
be subalgebras of it containing its unit and such that
for any
. Then there is a homomorphism of
-algebras
such that
,
. For
to be an isomorphism it is necessary and sufficient that there is in
a basis over
which is also a basis of the right
-module
.
Tensor product of two matrices (by D.A. Suprunenko)
The tensor product, or Kronecker product (cf. Matrix multiplication), of two matrices and
is the matrix
![]() |
Here, is an
-matrix,
is a
-matrix and
is an
-matrix over an associative commutative ring
with a unit.
Properties of the tensor product of matrices are:
![]() |
![]() |
![]() |
where ,
![]() |
If and
, then
![]() |
Let be a field,
and
. Then
is similar to
, and
, where
is the unit matrix, coincides with the resultant of the characteristic polynomials of
and
.
If and
are homomorphisms of unitary free finitely-generated
-modules and
and
are their matrices in certain bases, then
is the matrix of the homomorphism
in the basis consisting of the tensor products of the basis vectors.
Tensor product of two representations (by A.I. Shtern)
The tensor product of two representations and
of a group
in vector spaces
and
, respectively, is the representation
of
in
uniquely defined by the condition
![]() | (*) |
for all ,
and
. If
and
are continuous unitary representations of a topological group
in Hilbert spaces
and
, respectively, then the operators
,
, in the vector space
admit a unique extension by continuity to continuous linear operators
,
, in the Hilbert space
(being the completion of the space
with respect to the scalar product defined by the formula
) and the mapping
,
, is a continuous unitary representation of the group
in the Hilbert space
, called the tensor product of the unitary representations
and
. The representations
and
are equivalent (unitarily equivalent if
and
are unitary). The operation of tensor multiplication can be defined also for continuous representations of a topological group in topological vector spaces of a general form.
Comments
If is a representation of an algebra
in a vector space
,
, one defines the tensor product
, which is a representation of
in
, by
![]() |
In case is a bi-algebra (cf. Hopf algebra), composition of this representation with the comultiplication
(which is an algebra homomorphism) yields a new representation of
, (also) called the tensor product.
In case is a group, a representation of
is the same as a representation of the group algebra
of
, which is a bi-algebra, so that the previous construction applies, giving the same definition as (*) above. (The comultiplication on
is given by
.)
In case is a Lie algebra, a representation of
is the same as a representation of its universal enveloping algebra,
, which is also a bi-algebra (with comultiplication defined by
,
). This permits one to define the tensor product of two representations of a Lie algebra:
![]() |
Tensor product of two vector bundles
The tensor product of two vector bundles and
over a topological space
is the vector bundle
over
whose fibre at a point
is the tensor product of the fibres
. The tensor product can be defined as the bundle whose transfer function is the tensor product of the transfer functions of the bundles
and
in the same trivializing covering (see Tensor product of matrices, above).
Comments
For a vector bundle over a space
and a vector bundle
over a space
one defines the vector bundle
over
(sometimes written
) as the vector bundle over
with fibre
over
. Pulling back this bundle by the diagonal mapping
defines the tensor product defined above.
References
[1] | N. Bourbaki, "Elements of mathematics. Algebra: Algebraic structures. Linear algebra" , 1 , Addison-Wesley (1974) pp. Chapt.1;2 (Translated from French) |
[2] | F. Kasch, "Modules and rings" , Acad. Press (1982) (Translated from German) |
[3] | A.I. Kostrikin, Yu.I. Manin, "Linear algebra and geometry" , Gordon & Breach (1989) (Translated from Russian) |
[4] | P.R. Halmos, "Finite-dimensional vector spaces" , v. Nostrand (1958) |
[5] | M.F. Atiyah, "$K$-theory: lectures" , Benjamin (1967) |
Tensor product. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Tensor_product&oldid=38999